1. Frank Cowell: Risk Taking RISK TAKING MICROECONOMICS Principles and Analysis Frank Cowell Almost essential Risk Almost essential Risk Prerequisites March 2012 1

2. Frank Cowell: Risk Taking Economics of risk taking  In the presentation Risk we examined the meaning of risk comparisons in terms of individual utility related to people’s wealth or income (ARA, RRA)  In this presentation we put to this concept to work  We examine: Trade under uncertainty A model of asset-holding The basis of insurance March 2012 2

3. Frank Cowell: Risk Taking Overview… Trade and equilibrium Individual optimisation The portfolio problem Risk Taking Extending the exchange economy March 2012 3

4. Frank Cowell: Risk Taking Trade  Consider trade in contingent goods  Requires contracts to be written ex ante  In principle we can just extend standard GE model  Use prices p i  : price of good i to be delivered in state   We need to impose restrictions of vNM utility  An example: Two persons, with differing subjective probabilities Two states-of the world Alf has all endowment in state BLUE Bill has all endowment in state RED March 2012 4

5. Frank Cowell: Risk Taking Contingent goods: equilibrium trade OaOa ObOb x RED a b x BLUE b a  RED – ____  BLUE  RED – ____  BLUE b b Contract curve  Certainty line for Alf  RED – ____  BLUE  RED – ____  BLUE a a  Alf's indifference curves  Certainty line for Bill  Bill's indifference curves  Endowment point  Equilibrium prices & allocation March 2012 5

6. Frank Cowell: Risk Taking Trade: problems  Do all these markets exist? If there are  states-of-the-world… …there are n  of contingent goods Could be a huge number  Consider introduction of financial assets  Take a particularly simple form of asset: a “contingent security” pays $1 if state  occurs  Can we use this to simplify the problem? March 2012 6

7. Frank Cowell: Risk Taking Financial markets?  The market for financial assets opens in the morning  Then the goods market is in the afternoon  Use standard results to establish that there is a competitive equilibrium  Instead of n  markets we now have n+   But there is an informational difficulty To do financial shopping you need information about the afternoon This means knowing the prices that there would be in each possible state of the world Has the scale of the problem really been reduced? March 2012 7

8. Frank Cowell: Risk Taking Overview… Trade and equilibrium Individual optimisation The portfolio problem Risk Taking Modelling the demand for financial assets March 2012 8

9. Frank Cowell: Risk Taking Individual optimisation  A convenient way of breaking down the problem  A model of financial assets  Crucial feature #1: the timing Financial shopping done in the “morning” This determines wealth once state  is realised Goods shopping done in the “afternoon” We will focus on the “morning”  Crucial feature #2: nature of initial wealth Is it risk-free? Is it stochastic?  Examine both cases March 2012 9

10. Frank Cowell: Risk Taking Interpretation 1: portfolio problem  You have a determinate (non-random) endowment y  You can keep it in one of two forms: Money – perfectly riskless Bonds – have rate of return r: you could gain or lose on each bond  If there are just two possible states-of-the-world: rº < 0 – corresponds to state BLUE r' > 0 – corresponds to state RED  Consider attainable set if you buy an amount  of bonds where 0 ≤  ≤ y   March 2012 10

11. Frank Cowell: Risk Taking Attainable set: safe and risky assets x BLUE x RED  P  P 0 y y _ _ _ A  Endowment  If all resources put into bonds  All these points belong to A  Can you sell bonds to others?  Can you borrow to buy bonds? unlikely to be points here  If loan shark willing to finance you [1+rº]y _ [1+r' ]y _ y+  r′, y+  r  _ _ [1+r′ ]y, [1+r  ]y _ _ March 2012 11

12. Frank Cowell: Risk Taking Interpretation 2: insurance problem  You are endowed with a risky prospect Value of wealth ex-ante is y 0 There is a risk of loss L If loss occurs then wealth is y 0 – L  You can purchase insurance against this risk of loss Cost of insurance is  In both states of the world ex-post wealth is y 0 –   Use the same type of diagram March 2012 12

13. Frank Cowell: Risk Taking Attainable set: insurance x BLUE x RED  P y y _ _ _ A  Endowment  Full insurance at premium   All these points belong to A  Can you overinsure?  Can you bet on your loss? unlikely to be points here  P 0 y 0 – L y 0  L –  partial insurance March 2012 13

14. Frank Cowell: Risk Taking A more general model?  We have considered only two assets  Take the case where there are m assets (“bonds”)  Bond j has a rate of return r j,  Stochastic, but with known distribution  Individual purchases an amount  j, March 2012 14

15. Frank Cowell: Risk Taking A 1 5 7 4 6 3 2 Consumer choice with a variety of financial assets x BLUE x RED  Payoff if all in cash  Payoff if all in bond 2  Payoff if all in bond 3, 4, 5,…  Possibilities from mixtures  Attainable set  The optimum 5 4  P* P*  only bonds 4 and 5 used at the optimum March 2012 15

16. Frank Cowell: Risk Taking Simplifying the financial asset problem  If there is a large number of financial assets many may be redundant which are redundant depends on tastes… … and on rates of return  In the case of #  = 2, a maximum of two assets are used in the optimum  So the two-asset model of consumer optimum may be a useful parable  Let’s look a little closer March 2012 16

17. Frank Cowell: Risk Taking Overview… Trade and equilibrium Individual optimisation The portfolio problem Risk Taking Safe and risky assets  comparative statics March 2012 17

18. Frank Cowell: Risk Taking The portfolio problem  We will look at the equilibrium of an individual risk-taker  Makes a choice between a safe and a risky asset “money” – safe, but return is 0 “bonds”– return r could be > 0 or < 0  Diagrammatic approach uses the two-state case  But in principle could have an arbitrary distribution of r… March 2012 18

19. Frank Cowell: Risk Taking Distribution of returns (general case) r f (r)  loss-making zone  the mean  plot density function of r ErEr March 2012 19

20. Frank Cowell: Risk Taking Problem and its solution  Agent has a given initial wealth y  If he purchases an amount  of bonds: Final wealth then is y = y –  +  [1+r] This becomes y = y +  r, a random variable  The agent chooses  to maximise E u(y +  r)  FOC is E ( ru y (y +  * r) ) = 0 for an interior solution where u y () =  u() /  y  * is the utility-maximising value of   But corner solutions may also make sense…      March 2012 20

21. Frank Cowell: Risk Taking A Consumer choice: safe and risky assets x BLUE x RED y y _ _ P* P*  P 0   Attainable set, portfolio problem _  P  Equilibrium -- playing safe  Equilibrium - "plunging"  Equilibrium - mixed portfolio March 2012 21

22. Frank Cowell: Risk Taking Results (1)  Will the agent take a risk?  Can we rule out playing safe?  Consider utility in the neighbourhood of  = 0   E u(y +  r)  ———— | = u y (y ) E r  |   u y is positive  So, if expected return on bonds is positive, agent will increase utility by moving away from  = 0   March 2012 22

23. Frank Cowell: Risk Taking Results (2)  Take the FOC for an interior solution  Examine the effect on  * of changing a parameter  For example differentiate E ( ru y (y +  * r) ) = 0 w.r.t. y  E ( ru yy (y +  * r) ) + E ( r 2 u yy (y +  * r) )  * /  y = 0   * – E (ru yy (y +  * r)) —— = ————————  y E (r 2 u yy (y +  * r))  Denominator is unambiguously negative  What of numerator?       March 2012 23

24. Frank Cowell: Risk Taking Risk aversion and wealth  To resolve ambiguity we need more structure  Assume Decreasing ARA  Theorem: If an individual has a vNM utility function with DARA and holds a positive amount of the risky asset then the amount invested in the risky asset will increase as initial wealth increases March 2012 24

25. Frank Cowell: Risk Taking A  Attainable set, portfolio problem An increase in endowment P* P* x BLUE x RED y y _ _  P **  o y+  _ _    DARA Preferences  Equilibrium  Increase in endowment  Locus of constant   New equilibrium try same method with a change in distribution March 2012 25

26. Frank Cowell: Risk Taking A rightward shift r f (r)   original density function  original mean  shift distribution by   Will this change increase risk taking? March 2012 26

27. Frank Cowell: Risk Taking x RED A A rightward shift in the distribution x BLUE y y _ _ P **  P* P*   P 0  o  Attainable set, portfolio problem _  P  DARA Preferences  Equilibrium  Change in distribution  Locus of constant   New equilibrium What if the distribution “spreads out”? March 2012 27

28. Frank Cowell: Risk Taking A An increase in spread x BLUE x RED y y _ _ P* P*   P 0   Attainable set, portfolio problem _  P  Preferences and equilibrium  Increase r′, reduce r  y+  * r′, y+  * r  _ _  P * stays put  So  must have reduced  You don’t need DARA for this March 2012 28

29. Frank Cowell: Risk Taking Risk-taking results: summary  If the expected return to risk-taking is positive, then the individual takes a risk  If the distribution “spreads out” then risk taking reduces  Given DARA, if wealth increases then risk-taking increases  Given DARA, if the distribution “shifts right” then risk-taking increases March 2012 29